Question: Find the sum of the first $7$ terms in the following geometric series. Do not round your answer. $1024+512+256+...$
Explanation: This formula gives the sum ${S_n}$ of the first $ n$ terms in the geometric series where the first term is $ a$ and the common ratio is $C r$ : ${S_n}=\dfrac{ a(1-C r^{ n})}{1-C r}$ We are given the value for $ n$. We are also given the first terms in the series, which tells us the values for, $ a$ and $C r$. After we find them, let's plug them in the formula. We are given that ${n=7}$. Since the series is $1024+512+256+...$, we can tell that ${a=1024}$, and $C{r=\dfrac12}$. ${S_n}=\dfrac{{1024}\left(1-\left(C{\dfrac12}\right)^{{7}}\right)}{1-\left(C{\dfrac12}\right)}$ Evaluating the expression in the calculator, we get that $S_n=2032$. In conclusion, the sum of the first $7$ terms in the series is $2032$.